Customer Reviews

Introduction to Elliptic Curves and Modular Forms (Graduate Texts in Mathematics)

5 star:		(5)
4 star:		(0)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

This is not an easy book---it's certainly intended for graduate students---but it's also clearly the best introduction to the mathematics that give rise to the Taniyama-Shimura conjecture and the ideas that underlie the Wiles proof of Fermat's Last Theorem, and as such it deserves a prominent place in the world's intellectual life. And you have to love a math text that has a beautiful frontispiece (a sketch by Fomenko whose mathematical meaning I am still trying to work out).

Koblitz is in his element with this text. Much like Daniel Marcus's Number Fields, Koblitz develops a ground work to begin the study of elliptic curves. Here he builds upon the ancient problem of congruent numbers to help develop motivation for an in depth study of elliptic curves and modular forms. Not for the faint of (mathematical) heart, but a truly brilliant piece added to the number theoretic literature. A must own for those interested in advanced ideas behind elliptic curves.

The book provides an excellent and readable introduction to elliptic curves, using the complex analysis approach. It gives detailed proof to many theorems, and the exercises for each chapter are wonderful. It is a good supplement for those who use another book by Koblitz: Algebraic Aspects of Cryptography (AAC). It is funny that Koblitz has copied and pasted some of the paragraphs from Introduction to Elliptic Curves and Modular Forms to AAC!

This is a very well thought out and accessible introduction to modular forms that paves he way to further study and research. While not as general as Miyake and others, it explains the fundamental concepts driving the subject in a down to earth fashion not seen in modern text. Plus it leads to Shimura's ground breaking work on half integer weight modular forms and Tunnell's work on the congruent number problem, and its relation to the Swinnerton-Dyer conjecture. Motivated by the congruent number problem it aims and gets there.

I carefully worked through most of Koblitz's book. It is well written and worth the time to study if you are interested in modular forms and elliptic curves.